Finite-dimensional decentralized control of semilinear dissipative parabolic PDEs

We present a decentralized controller construction for the global stabilization of a class of semilinear dissipative parabolic PDE that have a linear part with a finite number of unstable eigenvalues; and a nonlinear part that satisfies a linear bound in the inner product. The controller is constructed by analyzing a control Lyapunov function based on the infinite-dimensional dynamics and applying a finite-dimensional linear quadratic regulator (LQR) controller. We first summarize the controller derivation for the full information case. It is found that a sufficient condition for the controller to stabilize the system is a simple finite-dimensional matrix inequality. We apply the stability analysis and control design procedure to construct a static output feedback controller and a decentralized controller for systems with collocated sensors and actuators. As an example, the control designs axe applied to the stabilization of the Kuramoto-Sivashinsky equation